∫ (x^3-2x^2+1/sqrt[3]{x^2}) wrt x eching

$$\int ( x ^ { 3 } - 2 x ^ { 2 } + \frac { 1 } { \sqrt[ 3 ] { x ^ { 2 } } } ) d x$$

Baholash

$$\frac{x^{4}}{4}-\frac{2x^{3}}{3}+3\sqrt[3]{x}+С$$

Yechim qadamlari

Summani muddatma-muddat integratsiya qiling.

$$\int x^{3}\mathrm{d}x+\int -2x^{2}\mathrm{d}x+\int \frac{1}{x^{\frac{2}{3}}}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int x^{3}\mathrm{d}x-2\int x^{2}\mathrm{d}x+\int \frac{1}{x^{\frac{2}{3}}}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{3}\mathrm{d}x integralni \frac{x^{4}}{4} bilan almashtiring.

$$\frac{x^{4}}{4}-2\int x^{2}\mathrm{d}x+\int \frac{1}{x^{\frac{2}{3}}}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{2}\mathrm{d}x integralni \frac{x^{3}}{3} bilan almashtiring. -2 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

$$\frac{x^{4}}{4}-\frac{2x^{3}}{3}+\int \frac{1}{x^{\frac{2}{3}}}\mathrm{d}x$$

\frac{1}{x^{\frac{2}{3}}} ni x^{-\frac{2}{3}} sifatida qaytadan yozish. k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{-\frac{2}{3}}\mathrm{d}x integralni \frac{x^{\frac{1}{3}}}{\frac{1}{3}} bilan almashtiring. Soddalashtiring va eksponensialdan ildiz shaklga aylantiring.

$$\frac{x^{4}}{4}-\frac{2x^{3}}{3}+3\sqrt[3]{x}$$

Агар F\left(x\right)f\left(x\right) ning dastlabki holati boʻlsa, u holatda f\left(x\right) ning barcha dastlabki holatlari toʻplami F\left(x\right)+C tarafidan belgilanadi. Shu sababli natijaga C\in \mathrm{R} integrallash konstantasini qoʻshing.

$$\frac{x^{4}}{4}-\frac{2x^{3}}{3}+3\sqrt[3]{x}+С$$

x ga nisbatan hosilani topish

$$x^{3}-2x^{2}+\frac{1}{x^{\frac{2}{3}}}$$